➔ Cosmo Grant

August 23rd, 2019

➔ Cosmo Grant

Is it impossible to be fair?

Statistical prediction is increasingly pervasive in our lives. Can it be fair?

The Allegheny Family Screening Tool is a computer program that predicts whether a child will later have to be placed into foster care. It's been used in Allegheny County, Pennsylvania, since August 2016. When a child is referred to the county as at risk of abuse or neglect, the program analyzes administrative records and then outputs a score from 1 to 20, where a higher score represents a higher risk that the child will later have to be placed into foster care. Child welfare workers use the score to help them decide whether to investigate a case further.

Travel search engines like Kayak or Google Flights predict whether a flight will go up or down in price. Farecast, which launched in 2004 and was acquired by Microsoft a few years later, was the first to offer such a service. When you look up a flight, these search engines analyze price records and then predict whether the flight's price will go up or down over some time interval, perhaps along with a measure of confidence in the prediction. People use the predictions to help them decide when to buy a ticket.

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October 10th, 2018

Who cares about stopping rules?

Can you bias a coin?

Challenge: Take a coin out of your pocket. Unless you own some exotic currency, your coin is fair: it's equally likely to land heads as tails when flipped. Your challenge is to modify the coin somehow—by sticking putty on one side, say, or bending it—so that the coin becomes biased, one way or the other. Try it!

How should you check whether you managed to bias your coin? Well, it will surely involve flipping it repeatedly and observing the outcome, a sequence of h's and t's. That much is obvious. But what's not obvious is where to go from there. For one thing, any outcome whatsoever is consistent both with the coin's being fair and with its being biased. (After all, it's possible, even if not probable, for a fair coin to land heads every time you flip it, or a biased coin to land heads just as often as tails.) So no outcome is decisive. Worse than that, on the assumption that the coin is fair any two sequences of h's and t's (of the same length) are equally likely. So how could one sequence tell against the coin's being fair and another not?

We face problems like these whenever we need to evaluate a probabilistic hypothesis. Since probabilistic hypotheses come up everywhere—from polling to genetics, from climate change to drug testing, from sports analytics to statistical mechanics—the problems are pressing.

Enter significance testing, an extremely popular method of evaluating probabilistic hypotheses. Scientific journals are littered with reports of significance tests; almost any introductory statistics course will teach the method. It's so popular that the jargon of significance testing—null hypothesis, $p$-value, statistical significance—has entered common parlance.

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